The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define γ-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset-Kasprzyk (2008) and Kasprzyk-Nill (2012).

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We show (quasi-)polynomiality and reciprocity results for the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017), Billera-Jia-Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). This proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.

This report documents the experience generated by a pilot study of an online mathematics course offered to last and second last year high school students. The course counted with 109 participants from Chile, Colombia and Ecuador. The goal of the course is to support students in secondary education in learning mathematics with the help of a virtual platform, enabling them to continue with higher studies. As a main activity of the pilot study, the project team in Chile realized the online course during seven workshops, where the students working in a computer laboratory were observed and supervised. A main conclusion is that the generation of working knowledge on how to manage the learning process using an online platform is a learning issue by its own, with implications that should not be underestimated. Even though most students are familiar with social network applications, there is no ability transfer when it comes to deal with productivity tools. One reason is the dependence of abilities on the used device. Dealing with smartphones is different to using desktop computers. For example, some workshop participants did not manage to scroll down the screen in order to find an exit button to confirm that a task has been completed and can be send to evaluation. Another example is that some do not manage the login procedure for the simple reason that they did not remember the password. They are not used to deal with passwords since usually smartphone applications do not request them. The implication for the overall project was that the session supervisors needed to track technical details including the login procedure. One interesting observation made was that most of the course used to quit their work immediately after the break time bell. This behavior falsifies the supposition that individual and self-organized work might overcome the usual classroom conditioning that aims to minimize the time spend in classroom. The adaption to a more autonomous working style, where the students work self-guided within the virtual environment took in average two sessions. Towards the end of the course, the students turned out to be motivated and in their majority interested in online learning, recognizing the complementary support to traditional classroom teaching.

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky ( 2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017) and Billera-Jia-Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.